Motivated by the desire to construct large independent sets in random graphs, Karp and Sipser modified the usual greedy construction to yield an algorithm that outputs an independent set with a large cardinal called the Karp-Sipser core. When run on the Erdős-Rényi $G(n,c/n)$ random graph, this algorithm is optimal as long as $c < e$. We will present the proof of a physics conjecture of Bauer and Golinelli (2002) stating that at criticality, the size of the Karp-Sipser core is of order $n^{3/5}$. Along the way we shall highlight the similarities and differences with the usual greedy algorithm and the $k$-core algorithm.
Based on a joint work with Nicolas Curien and Thomas Budzinski.
Further Information
Part of the Oxford Discrete Maths and Probability Seminar, held via Zoom. Please see the seminar website for details.